In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces.[1] The precise number of near misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons. Some high symmetry near-misses are also symmetrohedra with some perfect regular polygon faces.
Examples
| Name Conway name | Image | Vertex configurations | V | E | F | F3 | F4 | F5 | F6 | F8 | F10 | F12 | Symmetry |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Truncated triangular bipyramid t4dP3 | ) | 2 (5.5.5) 12 (4.5.5) | 14 | 21 | 9 | 3 | 6 | Dih3 order 12 | |||||
| Truncated triakis tetrahedron t6kT | ) | 4 (5.5.5) 24 (5.5.6) | 28 | 42 | 16 | 12 | 4 | Td, [3,3] order 24 | |||||
| Pentahexagonal pyritoheptacontatetrahedron | ) | 60 | 132 | 74 | 56 | 12 | 6 | Th, [3+,4] order 24 | |||||
| Chamfered cube cC | ) | 24 (4.6.6) 8 (6.6.6) | 32 | 48 | 18 | 6 | 12 | Oh, [4,3] order 48 | |||||
| -- | ) | 12 (5.5.6) 6 (3.5.3.5) 12 (3.3.5.5) | 30 | 54 | 26 | 12 | 12 | 2 | D6h, [6,2] order 24 | ||||
| -- | ) | 6 (5.5.5) 9 (3.5.3.5) 12 (3.3.5.5) | 27 | 51 | 26 | 14 | 12 | D3h, [3,2] order 12 | |||||
| Tetrated dodecahedron | ) | 4 (5.5.5) 12 (3.5.3.5) 12 (3.3.5.5) | 28 | 54 | 28 | 16 | 12 | Td, [3,3] order 24 | |||||
| Chamfered dodecahedron cD | ) | 60 (5.6.6) 20 (6.6.6) | 80 | 120 | 42 | 12 | 30 | Ih, [5,3] order 120 | |||||
| Rectified truncated icosahedron atI | ) | 60 (3.5.3.6) 30 (3.6.3.6) | 90 | 180 | 92 | 60 | 12 | 20 | Ih, [5,3] order 120 | ||||
| Truncated truncated icosahedron ttI | ) | 120 (3.10.12) 60 (3.12.12) | 180 | 270 | 92 | 60 | 12 | 20 | Ih, [5,3] order 120 | ||||
| Expanded truncated icosahedron etI | ) | 60 (3.4.5.4) 120 (3.4.6.4) | 180 | 360 | 182 | 60 | 90 | 12 | 20 | Ih, [5,3] order 120 | |||
| Snub rectified truncated icosahedron stI | ) | 60 (3.3.3.3.5) 120 (3.3.3.3.6) | 180 | 450 | 272 | 240 | 12 | 20 | I, [5,3]+ order 60 |
Coplanar misses
Some failed Johnson solid candidates have coplanar faces. These polyhedra can be perturbed to become convex with faces that are arbitrarily close to regular polygons. These cases use 4.4.4.4 vertex figures of the square tiling, 3.3.3.3.3.3 vertex figure of the triangular tiling, as well as 60 degree rhombi divided double equilateral triangle faces, or a 60 degree trapezoid as three equilateral triangles. It is possible to take an infinite amount of distinct coplanar misses from sections of the cubic honeycomb (alternatively convex polycubes) or alternated cubic honeycomb, ignoring any obscured faces.
Examples: 3.3.3.3.3.3
Rhombic prism
Gyroelongated trigonal pyramid
Triangulated monorectified tetrahedron
Elongated octahedron
Tetratetrahedron, Triangulated tetrahedron
Augmented triangular cupola
Triangulated truncated triangular bipyramid
Edge-contracted icosahedron
Hexagonal prism
Hexagonal antiprism,
Gyroelongated hexagonal pyramid
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
4.4.4.4
Square icositetrahedron
(Cube)
)
3.4.6.4:
Hexagonal cupola
(Degenerate)
)
See also
- Platonic solid
- Semiregular polyhedron
- Archimedean solid
- Prism
- Antiprism
- Johnson solids
- Geodesic sphere
- Goldberg polyhedron
References
- ^ Kaplan, Craig S.; Hart, George W. (2001), "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons", Bridges: Mathematical Connections in Art, Music and Science (PDF).